 3.7.1: Exer. 12: Find (a) (b) (c) (d)
 3.7.2: Exer. 12: Find (a) (b) (c) (d)
 3.7.3: Exer. 38: Find (a) , , , and (b) the domain of , , and fg (c)
 3.7.4: Exer. 38: Find (a) , , , and (b) the domain of , , and fg (c)
 3.7.5: Exer. 38: Find (a) , , , and (b) the domain of , , and fg (c)
 3.7.6: Exer. 38: Find (a) , , , and (b) the domain of , , and fg (c)
 3.7.7: Exer. 38: Find (a) , , , and (b) the domain of , , and fg (c)
 3.7.8: Exer. 38: Find (a) , , , and (b) the domain of , , and fg (c)
 3.7.9: Exer. 910: Find (a) (b) (c) (d)
 3.7.10: Exer. 910: Find (a) (b) (c) (d)
 3.7.11: Exer. 1120: Find (a) (b) (c) (d)
 3.7.12: Exer. 1120: Find (a) (b) (c) (d)
 3.7.13: Exer. 1120: Find (a) (b) (c) (d)
 3.7.14: Exer. 1120: Find (a) (b) (c) (d)
 3.7.15: Exer. 1120: Find (a) (b) (c) (d)
 3.7.16: Exer. 1120: Find (a) (b) (c) (d)
 3.7.17: Exer. 1120: Find (a) (b) (c) (d)
 3.7.18: Exer. 1120: Find (a) (b) (c) (d)
 3.7.19: Exer. 1120: Find (a) (b) (c) (d)
 3.7.20: Exer. 1120: Find (a) (b) (c) (d)
 3.7.21: Exer. 2134: Find (a) and the domain of and (b) and the domain of .
 3.7.22: Exer. 2134: Find (a) and the domain of and (b) and the domain of .
 3.7.23: Exer. 2134: Find (a) and the domain of and (b) and the domain of .
 3.7.24: Exer. 2134: Find (a) and the domain of and (b) and the domain of .
 3.7.25: Exer. 2134: Find (a) and the domain of and (b) and the domain of .
 3.7.26: Exer. 2134: Find (a) and the domain of and (b) and the domain of .
 3.7.27: Exer. 2134: Find (a) and the domain of and (b) and the domain of .
 3.7.28: Exer. 2134: Find (a) and the domain of and (b) and the domain of .
 3.7.29: Exer. 2134: Find (a) and the domain of and (b) and the domain of .
 3.7.30: Exer. 2134: Find (a) and the domain of and (b) and the domain of .
 3.7.31: Exer. 2134: Find (a) and the domain of and (b) and the domain of .
 3.7.32: Exer. 2134: Find (a) and the domain of and (b) and the domain of .
 3.7.33: Exer. 2134: Find (a) and the domain of and (b) and the domain of .
 3.7.34: Exer. 2134: Find (a) and the domain of and (b) and the domain of .
 3.7.35: Exer. 3536: Solve the equation .
 3.7.36: Exer. 3536: Solve the equation .
 3.7.37: Several values of two functions f and g are listed in the following...
 3.7.38: Several values of two functions T and S are listed in the following...
 3.7.39: If and , find
 3.7.40: If and , find
 3.7.41: If and , find
 3.7.42: There is one function with domain that is both even and odd. Find t...
 3.7.43: Let the social security tax function SSTAX be defined as , where is...
 3.7.44: Let the function CHR be defined by , . Then let the function ORD be...
 3.7.45: A fire has started in a dry open field and is spreading in the form...
 3.7.46: A spherical balloon is being inflated at a rate of . Express its ra...
 3.7.47: The volume of a conical pile of sand is increasing at a rate of , a...
 3.7.48: The diagonal d of a cube is the distance between two opposite verti...
 3.7.49: A hotair balloon rises vertically from ground level as a rope atta...
 3.7.50: Refer to Exercise 76 of Section 3.4. Starting at the lowest point, ...
 3.7.51: Refer to Exercise 77 of Section 3.4. When the airplane is 500 feet ...
 3.7.52: A 100footlong cable of diameter 4 inches is submerged in seawater...
 3.7.53: Exer. 5360: Find a composite function form for y.
 3.7.54: Exer. 5360: Find a composite function form for y.
 3.7.55: Exer. 5360: Find a composite function form for y.
 3.7.56: Exer. 5360: Find a composite function form for y.
 3.7.57: Exer. 5360: Find a composite function form for y.
 3.7.58: Exer. 5360: Find a composite function form for y.
 3.7.59: Exer. 5360: Find a composite function form for y.
 3.7.60: Exer. 5360: Find a composite function form for y.
 3.7.61: If and , approximate . In order to avoid calculating a zero value f...
 3.7.62: If and , approximate f g1.12 f g1.12 f f 5.2 2 . gx
Solutions for Chapter 3.7: Operations on Functions
Full solutions for Algebra and Trigonometry with Analytic Geometry  12th Edition
ISBN: 9780495559719
Solutions for Chapter 3.7: Operations on Functions
Get Full SolutionsChapter 3.7: Operations on Functions includes 62 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 62 problems in chapter 3.7: Operations on Functions have been answered, more than 36760 students have viewed full stepbystep solutions from this chapter. Algebra and Trigonometry with Analytic Geometry was written by and is associated to the ISBN: 9780495559719. This textbook survival guide was created for the textbook: Algebra and Trigonometry with Analytic Geometry, edition: 12.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.